Problems & Puzzles:
Conjectures
Conjecture 55. n
= m^2 + p + d
Werner Sand sends the following
conjecture:
Each natural number n > 2 is the sum
of a square (>0) and a prime or the sum of a square, a prime, and 1.
n = m^2 + p + d,
d = 0 v 1.
We can formulate the Goldbach
Conjecture (even version) in this way: Each
natural number n > 3 is the sum of 2 primes and (0 or 1). This has the
same
structure as my conjecture (except for the fact that the "1" is
trivial).
But my conjecture is much stronger than the GC, because there are less
squares in an interval than primes (less density). E.g.: For n=10^6
there
are more than 5.000 Goldbach partitions but only 63 "Sand partitions".
The
conjecture has also a certain similarity to the four squares theorem
(each
positive integer is the sum of at most 4 squares), because p can be the
sum
of 2 squares, but it is stronger.
With growing values of n, the difference d=1 is becoming relatively
unimportant and increasingly rare, if standing alone (no d=0 for the
same
n). So we can say (short form of the conjecture): Each natural number
n>2 is exactly or nearly the sum of a square and a prime.
Q1. What can you say
about this conjecture?
Addendum by CR (d=0):
If you concentrate only in n=m^2+p, m>0,
then obviously every n=x^2>9 has no solution, except when n-m=1 & n+m is
prime.
What about solutions for non-square n
values? The following 35 non-square n values has no solution type n=m^2+p:
5 10 13 31 34 37 58 61 85 91 127 130 214
226 370 379 439 526 571 706 730 771 829 991 1255 1351 1414 1549 1906 2986
3319 3676 7549 9634 21679
As a matter of fact Crespi de Valldaura
conjectures that
21679=7*19*163, is "Almost certainly the largest non-square integer
which is not the sum of a square and a prime"
Q2. Is there a
largest numbers such as 21679?
Contributions came from Farideh
Firoozbakht & Patrick Capelle. Capelle's contribution is a small treatise of
the current status of the art of the issue, plenty of references.
***
Farideh wrote:
It seems that in the following statement,
" If you concentrate only in n=m^2+p, m>0, then
obviously every n=x^2>9 has
no solution, except when n-m=1 & n+m is prime."
you made a typo mistake. The correct statement is:
"..., except when x-m=1 & x+m
is prime." namely "..., when 2*n^(1/2)-1=2x-1 is prime.".
Answer to Q1: I think this conjecture is true. I
checked it for all square numbers up to
10^12 and for all non-square numbers up to 10^7.
Answer to Q2: I think the Crespi de Valldaura's
conjecture is true.
I checked the validity of this conjecture up to
n=10^7.
We can generalize this conjecture as follows:
For each natural number k, k>1 there exists a
nutural number f(k) such
that f(k) isn't of the form m^k & m^k+p where m>0
and p is a prime and
if n>f(k) and n isn't of the form n=m^k then n is
of the form m^k+p .
For example Crespi de Valldaura's conjecture says
f(2)=21679 and I guess that
f(3)=78526384.
***
Patrick wrote:
Many authors have been interested by the
numbers which are the sum of a square and a prime, and some
of them (Klaus Brockhaus, Richard K. Guy, Axel Harvey, Dean
Hickerson, Vladeta Jovovic, T.D. Noe, Mike Oakes, John
Robertson, Felice Russo, Robert G. Wilson, Allessandro
Zacagnini, Reinhard Zumkeller and others) have worked on
some particular cases (e.g., n prime) or generalizations
(e.g., replacement of the square m^2 by a
power) [1],[2],[3],[4],[5],[6]. The conjecture proposed by
Werner Sand is interesting and probably new. It held my
attention by its simplicity and the use of the
logical symbol of disjunction. It is difficult to understand
the origin of this disjunction (in case it corresponds to
the association of several conjectures/ideas). So, for this
analysis, starting with some observations already known
(there are an infinity of squares n = a^2 which are not the
sum of a square and a prime [7] ; there are an infinity of
numbers which are not the sum of a nonzero square and a
prime [8] ; see also the table of Noe [9], including the CR
list), i collected several ideas coming from the net to
transform them into conjectures (some of them are not new,
except their formulation) and comments.
Conjecture A :
Every natural number n > 21679 is a square or the sum
of a square and a prime number [10],[11].
In other words, every natural number n > 21679 is a
square, a prime number or the sum of a nonzero square and a prime
number.
It's an improvement of a well known conjecture of
Hardy and Littlewood (conjecture H : every large number is either a
square or the sum of a square and a prime) [10].
James Van Buskirk checked it up to n =
3000000000 [1],[10],[11].
Conjecture B :
Every natural number n > 21679 is a square or the
sum of a nonzero square and a prime number.
Note that the conjecture A becomes the conjecture
B by using the conjecture D (when p > 21679).
Conjecture C :
Every nonsquare number n > 21679 is the sum of a
nonzero square and a prime number.
Note that the conjecture C is equivalent to the
conjecture B.
Conjecture D :
Every prime number p > 7549 is the sum of a
nonzero square and a prime number [12].
It's an improvement of a conjecture of
Mike Oakes (conjecture C in [13] ).
Conjecture E :
Every natural number n > 21679,
different from a square a^2 such that 2a - 1 is
composite, is the sum of nonzero square and a prime
number.
Conjecture F :
Every square number a^2, such
that 2a - 1 is composite, can be expressed as
a^2 = m^2 + p + 1, with m > 0 and p prime.
Conjecture G :
Every square number a^2, such
that 2a - 1 is prime, can be expressed as a^2 =
c^2 + p = d^2 + q + 1, with c > 0, d > 0, p
prime, q prime.
It means that if a square is
the sum of a nonzero square and a prime
number, then it is the sum of a nonzero
square, a prime number and 1.
Conjecture H :
Every square number a^2 >
1 can be expressed as a^2 = m^2 + p + 1, m >
0, p prime.
The conjecture C does not mean that there is
no square which is sum of a nonzero square and a prime.
Among the values of n, we have in fact two kinds of
squares : the squares n = a^2 where 2a - 1 is prime (squares
that i call R) and the squares n = a^2 where 2a - 1 is
composite (squares that i call S). It's a consequence of
Hickerson's proof [7] : if a^2 = m^2 + p, with p prime, then
p = (a-m)(a+m) ; so a - m = 1 and a + m = p ; hence 2a -1 =
p is prime. If 2a - 1 is not prime, then two possibilities :
2a - 1 = 1 (so a = 1 and n = 1, but it cannot be accepted
because Werner Sand specified that n > 2) or 2a - 1 is
composite. In this last case, n = a^2 cannot be expressed as
the sum of a (nonzero) square and a prime number.
A square R is always the sum of a nonzero square and a
prime, but it's never the case for a square S. The
conjecture E is a consequence of this situation. Considering
the conjectures above, I suppose that Werner Sand added the
possibility d = 1 essentially for cases where n is a square
S. If the main idea behind his conjecture is true, then i
guess that every square S can be expressed as S = m^2 + p +
1 (conjecture F). But it doesn't imply that the squares R
don't have the same property ! In fact, the squares R can
also be expressed in the form m^2 + p + 1, which gives me
opportunity to propose the conjecture G. Hence, in the
formulation of Sand's conjecture, d = 0 or 1 should be
replaced by d = 0 and/or 1, because there are numbers n for
which d = 0 and d = 1. The conjecture H is a consequence of
the conjectures F and G. Are there finitely many nonsquare
numbers which cannot be expressed in the form m^2 + p + 1, m
> 0, p prime ? I wonder whether every sufficiently large
number can be expressed in this form.
References :
[2] A065376 :
Primes of the form p + k^2, p prime and k > 0 (Nov 3, 2001).
[3] A073770 : Primes
p not of the form q + s where q is prime and s > 0 is the smallest
square such that q + s is prime (Aug 8, 2002).
[7]
A014090 : Numbers that are
not the sum of a square and a prime.
[8] A064233 :
Numbers which are not the sum of a prime number and a nonzero
square.
***
Werner Sand wrote on Oct 29:
Thanks for Farideh
Firoozbakht's generalization and Patrick Capelle's detailed documentation.
I din't know de
Valldaura's conjecture (VC) or other investigation in the vicinity of the
subject. I came from the following consideration: The sum of at most 4
squares can represent the natural numbers (Lagrange). This theorem I tried
to narrow and I found that 2 squares and 1 prime are sufficient, even if the
prime is not the sum of 2 squares (Euler). Finally, I was able to reduce
one of the 2 squares to the numbers 0 and 1, and this was my conjecture. The
fact that for n=square is always n = k^2 = m^2+p, i.e. p = k^2-m^2 = (k+m)(k-m),
thus p can be prime if k-m = 1, was of course my first thought. But p = k+m
is not necessarily prime, i.e. a square is not necessarily the sum of a
square and a prime, e.g. k(64) = 8, m(64) = 7, k+m = 15. In these cases,
however, n-1 is the sum of a square and a prime (d=0, my conjecture), and
then of course d(n)=1 (our example: 63 = 4+59 = 16+47). Thus there cannot be
2 consecutive n with d=only 1. It was evident that there are nonsquare n
besides the squares which are not the sum of a square and a prime. However,
I considered their number to be infinite, because I didn't know the VC. In
my opinion, there is little sense to differ square from nonsquare n as long
as the VC is not proven. But it seems to me important that there are
infinitely many n with d(n)=0 and infinitely many n with d(n)=1. And of
course that there is no need for d>1.
As to the
formulations "d = 0 v 1" and "d = 0 or 1": v comes from Latin vel and is
defined as "either…or…or both". It is not disjunction as P. Capelle says,
but adjunction. As well in the
German mathematical diction "oder" means "entweder…oder…oder beide".
I guess this applies to the
English "or", too. So there is no reason to change the formulation. By the
way I would have preferred set symbols, but unfortunately this is not
possible.
Outlook: We can
continue the procedure of refining and strengthening and restrict m^2 to
p^2, i.e. square primes. A single additional value of d (d=2) allows this:
Conjecture:
Every natural
number n>5 is the sum of a square prime, a prime, and one of the values
0,1,2: n = p^2+q+d, d out of (0,1,2).
I checked this until
n=10^6. Comparison number of partitions of n=10^6:
Goldbach: 5.402
m^2+p+d: 63
p^2+q+d: 47
...
In order to complete F. Firoozbakht's generalization:
For each natural number k there is a natural number D such
that for each
natural number n>2 applies:
n = m^k+p+d, d of {0,1,2…D}, m
natural number, p prime number.
Some values:
k D
1 0
2 1
3 3
4 8
5 15
6 17
7 22
We can call D a "measure of uncertainty".
***
Farideh Firoozbakht
wrote (Nov 15):
Jaroslaw Wroblewski
kindly wrote his comment about my generalization
of conjecture 55:
"Looks reasonable to me if k is prime. Otherwise I would change
n=m^k to n=m^d, d being a divisor of k greater than 1."
So the generalized form of conjecture 55 can be true only for primes k,
and we must improve it in the following form.
For each prime k, there exists a nutural number f(k) such that f(k)
isn't of
the form m^k & m^k+p where m>0, p is a prime and if n>f(k) and n isn't
of the form n=m^k then n is of the form m^k+p .
***
Richard Chen wrote on
March 2022:
This set is
the union of A020495 (composites)
and A065377 (primes),
A020495 is conjectured to be finite with 21 terms (largest term is
21679), A065377 is conjectured to be finite with 15 terms (largest term
is 7549), thus this set is also conjectured to be finite, with 21+15=36
terms.
Although this set
currently has no OEIS sequences, there is a sequence A064233 in
OEIS for all numbers not of the form m^2+p with m>=1 and prime p,
including the square numbers, the square numbers in A064233 are A104275^2,
thus the sequence A064233 is infinite, but this set (which is the
nonsquares in A064233) is conjectured to be finite, also there is a
sequence A014090 in
OEIS for all numbers not of the form m^2+p with m>=0 and prime p,
i.e. 0 is counted as a square (like A020495), since the A014090 also
contain A104275^2 as a subsequence of squares, A014090 is also
infinite. Besides, there are subsequences of this set in OEIS: A317966 (numbers
== 1 or 2 mod 4) and A308516 (odd
composites). There is also OEIS sequence for the number of ways to
write n as m^2+p with p prime: A064272 (m>=1)
and A002471 (m>=0)
A related problem
of this problem is for odd numbers n not of the form 2*m^2+p with
m>=1 and prime p, which is the sequence A060003,
and the primes in A060003 (together with the "oddest" prime 2) are
the Stern primes A042978,
A060003 is also conjectured to be finite with 10 terms (largest term
is 5993). There is also OEIS sequence for the number of ways to
write n as 2*m^2+p with p prime: A143539 (m>=1)
and A046923 (m>=0).
There is a sequence A347567 for
the combine with these two problems, the even numbers in A347567 are
2*(the numbers in this set) together the number 6 (6 is the only
number requiring the even prime 2), i.e. the union of 2*A020495 and
2*A065377 and {6}, and the odd numbers in A347567 are the numbers in
A060003, since all of A020495, A065377, A060003 are conjectured to
be finite, with 21, 15, 10 terms, respectively, A347567 is also
conjectured to be finite, with 21+15+10+1=47 terms, and the largest
term is 2*21679=43358
There is also a
similar problem about the triangular numbers (rather than the square
numbers), the nontriangular numbers not of the form m*(m+1)/2+p with
m>=1 and prime p, this set is A255904,
the primes in A255904 are A065397,
and it is conjectured that 216 is the only composite in A255904,
A065397 is conjectured to be finite with 4 terms (largest term is
211), thus A255904 is also conjectured to be finite, with 4+1=5
terms, and there is a sequence A111908 in
OEIS for all numbers not of the form m*(m+1)/2+p with m>=1 and prime
p, including the triangular numbers, the triangular numbers in
A111908 are A138666*(A138666+1)/2,
thus the sequence A111908 is infinite, but this set (which is the
nontriangular numbers in A111908) is conjectured to be finite, also
there is a sequence A076768 in
OEIS for all numbers not of the form m*(m+1)/2+p with m>=0 and prime
p, i.e. 0 is counted as a triangular number, since the A076768 also
contain A138666*(A138666+1)/2 as a subsequence of squares, A076768
is also infinite. Although there is no OEIS sequence for the number
of ways to write n as m*(m+1)/2+p with p prime, A132399 is
the sequence for "p is 0 or prime" rather than "p is prime", but of
course there is no difference between these two sequences for
nontriangular numbers n.
A related problem
of this problem is for odd numbers n not of the form m*(m+1)+p with
m>=1 and prime p, and it is conjectured that 1 and 3 are the only
such odd numbers. There is also OEIS sequence for the number of ways
to write n as m*(m+1)+p with p prime: A144590 (m>=1).
There is a sequence A347568 for
the combine with these two problems, the even numbers in A347568 are
2*A255904 together the number 10 (10 is the only number requiring
the even prime 2), i.e. the union of (2*the set of composites in
A255904, conjectured to be {432}) and 2*A065397 and {10}, and the
odd numbers in A347568 are conjectured to be {1, 3}, since all of
these three sequences are conjectured to be finite, with 1, 4, 2
terms, respectively, A347568 is also conjectured to be finite, with
1+4+2+1=8 terms, and the largest term is 2*216=432
***
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